Having obtained the covariant form of the Riemann tensor in a Cosmological \(L_{n}\)  we then went on and computed some of the basic objects that appear in applications, such as the Ricci tensor, Co-Ricci tensor, Ricci scalar etc. With the help of these relations we then proceeded in computing the generalized Gauss-Bonnet density. As expected our results show that the presence of the full non-metricity makes it impossible to express the latter density as a total differential in general. However, we proved that by constraining slightly the geometry by setting one of the non-metricity degrees of freedom to zero, the generalized version of the Gauss-Bonnet density becomes indeed a total derivative. Note that it was known in the literature that for Cartan-Weyl geometries (i.e. arbitrary torsion but Weyl non-metricity) the latter is indeed always a total differential [1]}. Here we have shown that in a Cosmological setup, the Gauss-Bonnet density is a total differential for geometries for which the non-metricity is more general from Weyl. Of course our result holds true for a Cosmological setup but this should not lead us to the conclusion that this would be so for any background. Here we proved that this is so only for highly symmetric FLRW backgrounds. It would be interesting, however, to see under what circumstances the Gauss-Bonnet term we presented here is a total derivative for the more general family of spherically symmetric Metric-Affine geometries as derived in [2]}Possibly, it would also be interesting to find the Gauss-Bonnet density analogues for the symmetric teleparallel [3]}, [4]} and general teleparallel [5]} Theories as was found for metric teleparallelism in [6]}.. Finally, it would be interesting to find the most general structure of the non-Riemannian geometries for which (REF ) is a total derivative irrespective of the background.
